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Theorem trljat1 29259
Description: The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 28957? (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trljat.l  |-  .<_  =  ( le `  K )
trljat.j  |-  .\/  =  ( join `  K )
trljat.a  |-  A  =  ( Atoms `  K )
trljat.h  |-  H  =  ( LHyp `  K
)
trljat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trljat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trljat1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )

Proof of Theorem trljat1
StepHypRef Expression
1 trljat.l . . . 4  |-  .<_  =  ( le `  K )
2 trljat.j . . . 4  |-  .\/  =  ( join `  K )
3 eqid 2253 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
4 trljat.a . . . 4  |-  A  =  ( Atoms `  K )
5 trljat.h . . . 4  |-  H  =  ( LHyp `  K
)
6 trljat.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
7 trljat.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
81, 2, 3, 4, 5, 6, 7trlval2 29256 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
98oveq1d 5725 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( R `  F )  .\/  P )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
10 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
11 hllat 28457 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
13 simp3l 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
14 eqid 2253 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1514, 4atbase 28383 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1613, 15syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
1714, 5, 6, 7trlcl 29257 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
18173adant3 980 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  e.  (
Base `  K )
)
1914, 2latjcom 14009 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( R `  F )  e.  ( Base `  K
) )  ->  ( P  .\/  ( R `  F ) )  =  ( ( R `  F )  .\/  P
) )
2012, 16, 18, 19syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( ( R `  F
)  .\/  P )
)
2114, 5, 6ltrncl 29218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2216, 21syld3an3 1232 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2314, 2latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
2412, 16, 22, 23syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
25 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
2614, 5lhpbase 29091 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2725, 26syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
2814, 1, 2latlej1 14010 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  ( F `  P )
) )
2912, 16, 22, 28syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  .<_  ( P  .\/  ( F `
 P ) ) )
3014, 1, 2, 3, 4atmod2i1 28954 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  ( F `  P )
) )  ->  (
( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
3110, 13, 24, 27, 29, 30syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( P  .\/  ( F `  P )
) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
32 eqid 2253 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
331, 2, 32, 4, 5lhpjat1 29113 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
34333adant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  ( 1. `  K ) )
3534oveq2d 5726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( W  .\/  P
) )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) ) )
36 hlol 28455 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3710, 36syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
3814, 3, 32olm11 28321 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
3937, 24, 38syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( 1. `  K
) )  =  ( P  .\/  ( F `
 P ) ) )
4031, 35, 393eqtrrd 2290 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
419, 20, 403eqtr4d 2295 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   1.cp1 13988   Latclat 13995   OLcol 28268   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   trLctrl 29251
This theorem is referenced by:  trljat3  29261  trlval4  29281  trlval5  29282  cdlemc5  29288  cdlemk1  29924  cdlemk8  29931  cdlemki  29934  cdlemksv2  29940  cdlemk7  29941  cdlemk12  29943  cdlemk15  29948  cdlemk7u  29963  cdlemk12u  29965  cdlemk21N  29966  cdlemk20  29967  cdlemk22  29986  cdlemm10N  30212
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252
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